Malcolm and Ravi raced each other. The average of their maximum speeds was $260\text{ km/h}$. If doubled, Malcolm's maximum speed would be $80\text{ km/h}$ more than Ravi's maximum speed. What were Malcolm's and Ravi's maximum speeds? Malcolm's maximum speed was
Explanation: Let $x$ represent Malcolm's maximum speed and let $y$ represent Ravi's maximum speed. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that the average of Malcolm's and Ravi's maximum speeds was $\textit{260 km/h}$. How can we model this sentence algebraically? The average of the maximum speeds is modeled by $\dfrac{x+y}{2}$. Since that average was $260\text{ km/h}$, we get the following equation: $\dfrac{x+ y}{2} = 260$ Let's multiply both sides of this equation by $2$ to avoid fractions: $x+ y = 520$ We are also given that twice Malcolm's maximum speed was $\textit{80 km/h}$ more than Ravi's maximum speed. This can be expressed as: $2x=y+80$ Let's rewrite this equation so that it's solved for $y$ : $y = 2x-80$ Now that we have a system of two equations, we can go ahead and solve it! Let's substitute $ y={2x-80}$ into the first equation: $ \begin{aligned}x+ y &= 520\\\\ x+({2x-80})&=520\\\\ 3x &=600\\\\ x&=200\end{aligned}$ Now we can substitute $x = 200$ into $x+y=520$ and find that $y=320$. Recall that $x$ denotes Malcolm's maximum speed and $y$ denotes the Ravi's maximum speed. Therefore, Malcolm's maximum speed was $\textit{200 km/h}$ and Ravi's maximum speed was $\textit{320 km/h}$.